## Project Ideas

This page contains a list of ideas for math DRP projects. You could model an entire project after one of these ideas, but they may also be useful as inspiration for general topics you might study. If you don’t see your dream project on this list, don’t worry – you’re welcome to come up with something totally different to do instead!

Each suggested project has an associated text, with the assumption that the project would focus on some subset of the contents of that text. The texts have links containing publishing information about the text along with a review (which may or may not be useful to you, depending on the review and your level of background knowledge).

Our project ideas also have recommended background for participants pursuing that project. These recommendations are meant to help you get the most out of the project, but they are also flexible in many cases. If your heart is set on a topic that you don’t have all the recommended background for, your DRP mentor may be able to arrange a project that covers some of the missing background as well as material from the topic of interest.

For more project ideas, you can also check out our list of previous projects or look at similar pages from other DRP programs. Both of these other sources contain suggestions that require varying levels of advanced courses as background. By contrast, most of the suggestions on this list require relatively few advanced math courses.

**Topic:** dynamical systems

**Text:** Devaney, An Introduction to Chaotic Dynamical Systems

**Recommended background:** A course in multivariable calculus (e.g. Math 53) is recommended, but the project may be possible without this background.

**Description:** The study of dynamical systems is a very broad topic, encompassing the mathematics of modeling any system that evolves over time. An active field of mathematical study, dynamical systems also has applications to many different sciences, e.g. physics and biology. This project is an introduction to dynamical systems, with a focus on systems which are considered “chaotic.”

**Topic:** game theory and transfinite numbers

**Text:** John Conway, On Numbers and Games

**Recommended background:** Math 55 (or the equivalent) would be useful but is not necessary.

**Description:** The suggested text begins by describing a formal theory of infinite numbers. It then introduces game theory and relates games to infinite numbers in a few different interesting ways. Those more interested in game theory could probably skip some of the theory of infinite numbers to focus primarily on the game theory.

**Topic:** topology

**Text:** Jänich, Topology (translated by Silvio Levy)

**Recommended background:** None

**Description:** This project is an introduction to point-set topology, which is essentially the study of the “shape” of various spaces and objects (as opposed to geometry, which tends to be concerned with distances and sizes as well as shapes). It begins from the definition of a topology and explores several different important topics in the subject. While these topics would certainly be useful to someone interested in topology, they are also important to many other areas of math, including analysis and geometry.

**Topic:** differential geometry

**Text:** Do Carmo, Differential Forms and Applications

**Recommended background:** A course in multivariable calculus (e.g. Math 53) is recommended.

**Description:** In multivariable calculus, one learns how to take derivatives and integrals of real-valued functions in multiple variables. However, these functions always live in two- or three-dimensional real space (i.e. R^2 or R^3). The idea of differential geometry is to generalize multivariable calculus in order to take derivatives and integrals of functions on many different kinds of spaces, such as curved surfaces in R^3. This project provides an introduction to differential geometry from the perspective of multivariable calculus.

**Topic:** geometry and algebraic topology

**Text:** Madsen and Tornehave, From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes

**Recommended background:** Courses in multivariable calculus (e.g. Math 53) and linear algebra (e.g. Math 54) are recommended. Some knowledge of topological spaces would be useful but is not required.

**Description:** Cohomology is a ubiquitous tool in modern mathematics – almost every active field of study utilizes some type of cohomology to some extent. De Rham cohomology is an ideal type of cohomology to learn first, and it plays a very important role in geometry. This project will provides an introduction to De Rham cohomology while assuming no prior knowledge of either geometry or cohomology. The project will also introduce some foundations of geometry and explore how De Rham cohomology is used in that subject.

**Topic:** complex analysis

**Text:** Tristan Needham, Visual Complex Analysis

**Recommended background:** A course in multivariable calculus (e.g. Math 53) is recommended. Math 104 would be useful but is not required.

**Description:** A project using this text would most likely be an introduction to complex analysis. Although the text covers much of the material in Math 185, it gives a very visual, geometric treatment of complex analysis that one would probably not see in that class. There are also several sections on interpretations by and applications to physics, which a project could focus on either a lot or very little depending on interest.

**Topic:** applications of number theory

**Text:** Conway and Fung, The Sensual Quadratic Form

**Recommended background:** None

**Description:** The suggested text focuses on quadratic forms and also touches on p-adic numbers, two important topics in algebraic number theory. However, the goal is not to thoroughly understand these topics but rather to understand their applications to certain physical questions related to the sensory perception of humans. This makes for a visual, geometric, and accessible approach to the math involved.

**Topic:** number theory and computer science

**Text:** Davenport, The Higher Arithmetic

**Recommended background:** Math 55 (or the equivalent) would be useful but is not necessary.

**Description:** This project introduces several important classical topics in number theory, such as quadratic residues and Diophantine equations. It then explores the intersection of number theory and computer science in the form of prime factorization algorithms and cryptography.

**Topic:** number theory

**Text:** Hardy and Wright, Introduction to the Theory of Numbers

**Recommended background:** Math 55 would be helpful but is not required.

**Description:** These days, number theory typically involves a lot of abstract algebra (for the language of field extensions, groups, etc.) The suggested text introduces many major topics of number theory without assuming any knowledge of abstract algebra, making it relatively accessible for most DRP participants. This project would likely cherry-pick from the many varied chapters in the text according to the participant’s interest.

**Topic:** algebraic number theory (specifically, class field theory)

**Text:** David Cox, Primes of the Form x^2+ny^2

**Recommended background:** A course in abstract algebra (e.g. Math 113) is recommended.

**Description:** Class field theory, roughly speaking, is the study of field extensions of the rational numbers with abelian Galois group. This project is an historical and example-motivated introduction to class field theory. The suggested text focuses on one example problem in particular: given a fixed integer n, can we characterize prime numbers that can be written as x^2+ny^2 for some integers x and y? The text assumes no background in number theory, but those that do have a background in the subject could skip the earlier chapters and focus more on the discussion of modular forms and/or elliptic curves in Chapter 3.

**Topic:** number theory

**Text:** Ireland and Rosen, A Classical Introduction to Modern Number Theory

**Recommended background:** A course in abstract algebra (e.g. Math 113) is recommended.

**Description:** This project provides an introduction to number theory by studying many of the important problems and examples that have historically motivated the field. It builds up from the foundations of number theory to key topics in the modern understanding of analytic and/or algebraic number theory. The project could focus on many different such topics depending on interest.

**Topic:** algebraic geometry and number theory

**Text:** Silverman and Tate, Rational Points on Elliptic Curves

**Recommended backgroun**d: Knowledge of group theory (e.g. from Math 113) is recommended. Knowledge of field theory (e.g. from Math 113) would be helpful but is not required. A student taking Math 113 concurrently would probably have sufficient background.

**Description:** Elliptic curves are equations of the form y^2 = x^3 + ax + b for some fixed complex numbers a and b. (More generally, one can take a and b to lie in any field.) Elliptic curves turn out to have a very rich structure, which makes them a foundational subject in both algebraic geometry and number theory. This project is an introduction to elliptic curves that assumes no background in either algebraic geometry or number theory. It would be ideal for students who either want to try out one of these subjects or want to start building knowledge of these subjects. The project could also focus on applications of elliptic curves to cryptography, depending on interest.